[unpaidmoon]

Change and mathematical scope

Scoped counting proof for one proposition in Cinna Davis's PR CrocSwap#4100 (CRT.md §2 and the displayed guard comparison in §4). No new κ, and the selected conditional exponent is unchanged.

For every integer p >= 2 and ell >= 1, a sequential evaluator that reconstructs one CRT group of bit length at most p and builds four rotation-switch networks for a batch of at most 8(p+2) axes uses at most

E(p, ell) = 2^{4 ell} (p+2)^{16}

bit-steps. The constant in CrocSwap#4100's displayed bound may therefore be C = 1 on this domain. The same bound is exactly

2^{-G} E(p, ell) <= 2^{-60 ell} (p+2)^{-48},

with G = 64 (ell + ceil(log2(p+2))).

CrocSwap#4100's size paragraph gives group bit length at most p for n >= 4 (log2 T <= b + 1 and nominal p = 6b). The factor 8 is this package's explicit reading of "at most O(p) axis controls", not a new parameter of that construction. The bit-step count is a counting argument. The executed checker uses ordinary arrays and is not a finite-state tape trace.

Not proved: reservoir placement, the ordinary interchange callee, sparse repair, native execution, or the outer multiplication theorem.

Task and overlap

• Task: uniform-transfer-audit, scope refined to this one evaluator proposition.
• Base: canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244, priority revision 7.
• Propose unconditional κ = 0.00173097911567079247276057 via batched CRT CrocSwap/integer-mult-bounds#4100 head cited by the family record: 337d04939571a637aeab54a0f58f54fde811663a.
• Ordered eligible list used for this session, after dropping obligations that already have an open audit: the coverage map's shortest unaudited list for Propose unconditional κ = 0.00173097911567079247276057 via batched CRT CrocSwap/integer-mult-bounds#4100 (PR Map independent audit coverage of #4100's obligations (navigation; no new κ) CrocSwap/integer-mult-bounds#4311). Item 4 is this evaluator. The other four are BIT fixed tapes, the 2^{-4p} constant, Gaussian row-field movement, and the arbitrary-f complete cover.
• PR Audit uniform Beneš compiler with charged sequential arena scans CrocSwap/integer-mult-bounds#4192 prices the Beneš component only after the permutation is encoded, and leaves CRT reconstruction open. This package does not depend on its unmerged bound.
• PR Audit conditional CRT last-K two-slice streaming tariff CrocSwap/integer-mult-bounds#4187 assumes the uniform evaluator and prices the last-K branch.
• PR Audit the compact-switch coefficient and guard contract. CrocSwap/integer-mult-bounds#4321 compares a guard-density window with the printed 2^{-G} E display and does not derive E.
• PR Audit CRT reservoir placement and split-axis coordinate descriptors CrocSwap/integer-mult-bounds#4174 prices reservoir placement.

Validation

Python 3.12, standard library only, from the repository root:

python3 scripts/research_package.py --package research/crt-uniform-evaluator --check-only
python3.12 -B research/crt-uniform-evaluator/verify.py
python3 scripts/check_research_commit.py --package research/crt-uniform-evaluator --check-source-history

verify.py prints PASS_CRT_UNIFORM_EVALUATOR. It checks the integer comparisons, the Beneš scan recurrence, schoolbook and division tariffs, 90 agreements with the retained benes compiler, 44 CRT offset fixtures, and rejection of a non-invertible prefix and a non-permutation. Metadata pins and the source-history preflight passed locally. No Lean, Docker, or full CrocSwap#4100 replay was run.

Credit

Cinna Davis supplies the CRT construction, the E(p, ell) shape, and the retained compiler in PR CrocSwap#4100. ToikY's PR CrocSwap#4192 is the predecessor Beneš-scan audit and names this follow-up. PR CrocSwap#4311 records that the evaluator was still open. Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0. No independent human review or proof-assistant acceptance is claimed.

Open next questions

1. Does one root-axis query still fit the same bound when every logical access scans a single shared arena, as in PR Audit uniform Beneš compiler with charged sequential arena scans CrocSwap/integer-mult-bounds#4192, after this CRT offset is included? First step: encode the offset as that arena's permutation and compare touch counts. Blocked because Audit uniform Beneš compiler with charged sequential arena scans CrocSwap/integer-mult-bounds#4192 starts from an encoded permutation and is not replayed here.
2. With C = 1, do reservoir placement and the ordinary interchange callee in CRT.md §3–§4 still fit their stated bills, including the finite elementary fallback? First step: check H >= G on Propose unconditional κ = 0.00173097911567079247276057 via batched CRT CrocSwap/integer-mult-bounds#4100's size family. Blocked because placement geometry is PR Audit CRT reservoir placement and split-axis coordinate descriptors CrocSwap/integer-mult-bounds#4174's obligation.

Workflow feedback (optional)

The CrocSwap#4311 coverage map made the remaining CrocSwap#4100 obligations enumerable. A structured audits field on TASK.json would have made that map unnecessary.